Anisotropic Complex Refractive Indices of Atomically Thin Materials: Determination of the Optical Constants of Few-Layer Black Phosphorus

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Article Anisotropic Complex Refractive Indices of Atomically Thin Materials: Determination of the Optical Constants of Few-Layer Black Phosphorus

Aaron M. Ross 1, Giuseppe M. Paternò 2 , Stefano Dal Conte 1, Francesco Scotognella 1,2,* and Eugenio Cinquanta 3 1 Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy; [email protected] (A.M.R.); [email protected] (S.D.C.) 2 Center for Nano Science and Technology@PoliMi, Istituto Italiano di Tecnologia (IIT), Via Giovanni Pascoli, 70/3, 20133 Milan, Italy; [email protected] 3 Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, 20133 Milan, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-02-2399-6056

Received: 5 November 2020; Accepted: 14 December 2020; Published: 16 December 2020

Abstract: In this work, studies of the optical constants of monolayer transition metal dichalcogenides and few-layer black phosphorus are briefly reviewed, with particular emphasis on the complex dielectric function and refractive index. Specifically, an estimate of the complex index of refraction of phosphorene and few-layer black phosphorus is given. The complex index of refraction of this material was extracted from differential reflectance data reported in the literature by employing a constrained Kramers–Kronig analysis combined with the transfer matrix method. The reflectance contrast of 1–3 layers of black phosphorus on a silicon dioxide/silicon substrate was then calculated using the extracted complex indices of refraction.

Keywords: transition metal dichalcogenides; black phosphorus; two-dimensional materials; complex refractive index; Kramers–Kronig analysis

1. Introduction Two-dimensional materials emerged as a promising option for the development of nanoelectronic and optoelectronic devices, due to their peculiar electronic and optical properties. In this respect, transition metal dichalcogenides (TMDs), when thinned to a single atomic layer, show an indirect–direct band gap transition, inequivalent valleys in the Brillouin zone and non-trivial topological order [1–3]. In the vast family of two-dimensional materials, elemental ones represent a niche, due to their high reactivity at ambient conditions, that makes their characterization and exploitation challenging [4,5]. Among them, phosphorene, i.e., monolayer black phosphorous (BP), is the most promising material for optoelectronic and photonic applications, as its electronic band gap lies in a spectral region between the ones of TMDs and the one of graphene [6]. For an effective employment of these materials in photonic and optoelectronic devices, it is important to determine their dielectric response. Among the monolayers of transition metal dichalcogenides, MoS2 was the first material on which optical measurements were performed. In fact, in 2013 and 2014, the complex dielectric function of MoS2 was measured with spectroscopic ellipsometry by Shen et al. [7], Yim et al. [8] and Li et al. [9]. The complex dielectric functions of monolayers MoS2, WS2, MoSe2 and WSe2 were measured by Li et al. by employing reflectance contrast spectroscopy [10]. In 2019, the monolayers, together with the bilayers and the trilayers, of the four above mentioned chalcogenides were studied by Hsu et al. [11]. In 2020, Ermolaev et al. [12] measured the complex dielectric function of MoS2 in a broad range of

Materials 2020, 13, 5736; doi:10.3390/ma13245736 www.mdpi.com/journal/materials Materials 2020, 13, 5736 2 of 12 wavelengths, i.e., 290–3300 nm. Ermolaev et al. [13] also measured the complex index of refraction of WS2 in the range of 375–1700 nm. Additionally, a number of studies investigated the anisotropic components of TMD and phosphorous few-layer systems via the incorporation of dielectric and plasmonic nanostructures [14–21]. In one study, the coupling of electromagnetic fields with the out-of-plane component of the MoS2 dielectric function was possible due to the peculiar morphology of chemical vapor deposition (CVD)-grown few-layer MoS2 on rippled substrate, hence promoting 1D nanostructures as a promising path for the full exploitation of the few-layer MoS2 dielectric response [19]. Similarly, a top-down approach was revealed to be effective for the strain-induced modification of the electronic band structure of atomically thin MoS2 and black phosphorous, paving the way for the on-demand tuning of their optical properties [20,21]. One study investigated exfoliated MoS2 on a lithographically defined SiO2 nanocone substrate, demonstrating a deterministic red-shift of the A exciton photoluminescence (PL) due to elastic strain [14]. Another study utilized spatially and time-resolved PL diffusion measurements of WSe2 exfoliated onto a 1.5 µm diameter SiO2 pillar, resulting in a similar red-shift and demonstration of exciton diffusion towards high tensile strain regions; that work was a proof of concept for engineerable excitonic diffusion that may lead to a new generation of opto-excitonic devices [15]. Strain engineering may also be used to enable new radiative pathways for the optical excitation of selection rule forbidden TMD dark excitons under normal incidence conditions; these dark and “gray” excitons possess much longer radiative lifetimes (>100 ps) [16,22,23]. In the absence of intentional strain, one study utilized a high numerical aperture objective lens (NA = 0.82) to characterize the dark exciton in WSe2: even for normal incidence, a Gaussian beam with a tight focus has a non-zero electric field component along the beam propagation direction [16]. However, optical excitation of the dark exciton in WSe2 was more easily enabled by coupling to the surface plasmon polariton of silver (Ag) [17]. Another study incorporated a hexagonal boron nitride (hBN)-encapsulated WSe2 monolayer into a plasmonic modulator device, requiring precise knowledge of the TMD, gold (Au) and hBN complex indices of refraction [18]. It thus emerges how the development of simple and robust tools for the careful analysis of the dispersion of the complex refractive index is beneficial for an appropriate engineering of devices that include two-dimensional materials. In contrast, phosphorene (single-layer BP) and few-layer BP have not been studied with the same fervor as their TMD counterparts, likely due to their relative sensitivity to degradation. One significant distinction between TMDs and BP is that the band structure of BP exhibits considerable anisotropy: that is, the effective electron mass along the armchair and zig-zag directions is a factor of five times larger in the former direction [24]. This anisotropy is manifest in polarization-sensitive absorption and PL measurements [24–27], as well as spatially resolved excitonic diffusion measurements [24]. The relatively fragile phosphorene system holds considerable promise due to the ease of PL/absorption tunability by layer number tuning, which can shift the resonance energies between 0.32 and 1.7 eV [24]. We focus on estimating the complex refractive index of phosphorene, which shows a morphological in-plane anisotropy resulting in an anisotropic optical response [25]. We retrieve the refractive index along the armchair (AC) and zig-zag (ZZ) direction by exploiting the constrained Kramers–Kronig analysis (CKKA) and fit experimental contrast reflectivity with a model obtained by means of the transfer matrix method (TMM).

2. Materials and Methods Methodology for extraction of the complex index of refraction of few-layer BP from reflectance contrast data: The linear optical properties of thin films are commonly measured via static absorption or reflectivity contrast measurements. Experimental techniques include normal incidence reflectivity contrast (dR/R) over a wide plane of view, spatially resolved confocal dR/R in the case of exfoliated small-area TMDs, few-layer BP, graphene and graphene oxide and simultaneous absolute reflectivity and transmission measurements for the determination of static absorption [10,11,25,27–30]. Although these methods reveal the static absorption A and reflectivity dR/R and transmission dT/T contrasts of the thin films Materials 2020, 13, x 5736 FOR PEER REVIEW 3 of 12 these methods reveal the static absorption A and reflectivity dR/R and transmission dT/T contrasts ofrelative the thin to the films linear relative substrate to the response, linear substrate the determination response, of the the determination complex index of of the refraction complexen =indexn + ofik, = + i n = √ refractionor equivalently 𝑛 = the𝑛+𝑖𝑘 complex, or equivalently dielectric function the complexε dielectricεr εi, where function 𝜀 ε =, typically𝜀 +𝑖𝜀, where requires 𝑛 more = √𝜀, typicallysophisticated requires optical more characterization sophisticated methodsoptical char suchacterization as ellipsometry methods [8,31 such–33]. as ellipsometry [8,31– 33]. In this section, we extract the complex index of refraction of hBN-encapsulated few-layer BP on a sapphireIn this substratesection, we from extract confocal the complex reflectivity index contrast of refraction data taken of hBN-encapsulated from [25]. Specifically, few-layer the layered BP on sample structure is as follows (see Figure1): air 15 nm hBN few-layer BP α-Al O (sapphire); a sapphire substrate from confocal reflectivity contrast→ data taken→ from [25]. Specifically,→ 2 3 the layered sampleBP thicknesses structure range is as follows nominally (see fromFigure 0.5 1): nmair  (1L) 15 nm to 2.5 hBN nm  (5L),few-layer and bulkBP BP 𝛼-Al is2 100O3 (sapphire); nm thick. WeBP thicknesses assume that range the exfoliated nominally hBN from layer 0.5 is nm crystalline, (1L) to 2.5 rather nmthan (5L), isotropic and bulk as BP for is the 100 case nm of thick. cubic We BN assumeor polycrystalline that the exfoliated hBN [33– hBN35] with layer an is in-plane crystalline, dielectric rather constant than isotropic given byas for the case of cubic BN or polycrystalline hBN [33–35] with an in-plane dielectric constant given2 by 3.336 λ ε = n2 = 1 +3.336 𝜆 (1)(1) hBN, hBN, 2 𝜀,∥ =k 𝑛,∥ = 1+ λ 26322 k 𝜆 − 26322− where the wavelength λ is given in nanometers [33]. Additionally, we also use a Sellmeier equation to where the wavelength 𝜆 is given in nanometers [33]. Additionally, we also use a Sellmeier equation describe the α-Al2O3 substrate index of refraction [36]. to describe the 𝛼-Al2O3 substrate index of refraction [36].

Figure 1. SchematicSchematic diagram diagram of of multi-layer multi-layer system under study: air, air, 15 nm of hBN, hBN, 1–5 layers layers of black black phosphorus (BP) and a thick substrate of 𝛼 -Al2O3 [25]. Input and output wave are indicated phosphorus (BP) and a thick substrate of α-Al2O3 [25]. Input and output wave are indicated directionally directionallyby arrows, where by arrows, only a fewwhere reﬂected only a waves few reflecte from eachd waves interface from are each indicated. interface The are incident indicated. angle The is incidentshown as angle oblique is shown for clarity as oblique (normal for incidence clarity (normal in experiment). incidence in experiment).

One commonly used method for the determinationdetermination of the complex linear material response functions is thethe Kramers–KronigKramers–Kronig analysisanalysis (KKA): (KKA): it it is is well well known known that that the the real real and and imaginary imaginary parts parts of ofthe the dielectric dielectric function function are relatedare related to one to anotherone anot viaher the via Kramers–Kronig the Kramers–Kronig (KK) relations, (KK) relations, which imposewhich imposecausality causality conditions conditions on the responseon the response functions functions [37–39 ].[37–39]. However, However, the integrals the integrals involved involved in these in theserelations relations are taken are taken over inﬁniteover infinite limits; limits; practical practi implementationcal implementation requires requires the acquisition the acquisition of data of data over overa wide a wide energy energy range, range, and may and bemay limited be limited to materials to materials which which exhibit exhibit an optical an optical response response only over only a overnarrow a narrow wavelength wavelength range. range. We utilize utilize an an alternative alternative method method called called the constrained the constrained Kramers–KronigKramers–Kronig analysis analysis (CKKA), (CKKA), which waswhich introduced was introduced by Kuzmenko by Kuzmenko to address to address some of some the ofproblems the problems associated associated with KKA with KKA[37]. This [37]. methodThis method takes advantage takes advantage of the offact the that fact basis that functi basisons functions such as such the complex as the complex Lorentzian Lorentzian can be chosen can be thatchosen individually that individually satisfy the satisfy KK therelati KKons. relations. Such a function Such a function is given is by given by 𝜔 (2) 𝜀 = , ω2 𝜔, −𝜔 +𝑖𝜔𝛾p,k ε = (2) k 2 2 ω ω + iωγk 0,k − where 𝜔, is the plasma frequency, 𝜔, is the oscillator frequency and 𝛾is the linewidth. Since linear combinations2 of such basis functions2 also satisfy the KK relations, the method introduces a where ωp,k is the plasma frequency, ω0,k is the oscillator frequency and γk is the linewidth. Since linear meshcombinations in energy of space such of basis a large functions number also of satisfyoscillators the KK𝑁 relations,~𝑁; the the oscillator method freq introducesuencies aare mesh fixed, in andenergy the space linewidth of a large is chosen number as ofthe oscillators spacing betweenNosc~Ndata oscillator; the oscillator energies. frequencies Thus, the are only ﬁxed, remaining and the parameterslinewidth is left chosen to minimize as the spacing are the betweenoscillator oscillator strengths energies. (or plasma Thus, frequencies): the only remaining hence, the parametersconstrained KK analysis. To further optimize the CKKA fitting procedure, we chose functions that are similar to

Materials 2020, 13, 5736 4 of 12 left to minimize are the oscillator strengths (or plasma frequencies): hence, the constrained KK analysis. To further optimize the CKKA fitting procedure, we chose functions that are similar to the complex Lorentzian of Equation (2) and also satisfy the KK relations, but are more locally weighted, avoiding the contributions of the tails of the Lorentzian distributions far away from the central oscillator energy. One such function is a triangular function for εi, with a corresponding analytically integrable εr [37,38]. We note that although this method can easily reproduce reflectivity contrast data when a sufficiently large oscillator mesh is used, it is sensitive to a lack of information about the optical response outside of the experimental acquisition range [10,37,38]. This problem is not entirely unexpected, considering that it is fundamentally a KK-type analysis method. Often, high-frequency information about a given optically active system is known via X-ray photoelectron emission spectroscopy or UV absorption experiments; additionally, low-frequency information is often revealed via Fourier transform IR (FTIR) and Raman scattering experiments [5,40–42]. However, such information is currently lacking for few-layer BP, although bulk BP is well characterized. Although similar calculations using the CKKA method have extracted the complex index of refraction for TMD monolayer and few-layer systems assuming a bulk-like response at high energies [10], no such assumptions are made here for the few-layer BP fitting procedure. For a given complex index of refraction, the optical response of a multi-layered system is then determined by the transfer matrix method (TMM) [43]. It is noted that perturbative “linearized” methods have also been developed for the treatment of TMD monolayers, few-layer BP and graphene [10,25,30], which treat the optical response of the thin film using the sheet conductivity

σd = σd = idω(ε 1). (3) − − For basic layered systems such as air monolayer TMD SiO , relatively simple forms for the → → 2 reflectivity contrast dR/R can be derived [29]. However, for the system under study in this section (air hBN BP α-Al O ), with no assumptions being made about reflection coefficients between → → → 2 3 interfaces, no tractable analytical form for dR/R is derived; note that this stands in contrast to previous studies [25] which assume that the BP α-Al O interface does not contribute to the overall optical → 2 3 response of the layered system. We use the standard TMM [43] to relate the input and reflected fields at the air hBN interface → (1) to the transmitted fields at the BP α-Al O interface (3). Specifically, for normal incidence fields, → 2 3 the electric and magnetic fields, which are implicit sums of all fields (incident, reflected, transmitted) are given by

" # " #  i  E E  cos φij sin φij  2πn0nijdijE 1 = Mˆ Mˆ 3 Mˆ =  nij  = 1 2 2 3 , i j  , φij (4) H1 − − H3 − inij sin φij cos φij hc which can then be related to the field reflection coefficient by the following relation

Er n m + n nsm m nsm r = = 0 11 0 12 − 21 − 22 (5) Ei n0m11 + n0nsm12 + m21 + nsm22 where mij is the ij-th element of the product of the two matrices in Equation (4), n0, ns are the indices of refraction of air and the back substrate (α-Al2O3), hc is the product of the speed of light and Planck’s hc constant, E = λ is energy, nij is the index of refraction for the material layer between the i and j interfaces and dij is the thickness between the i and j interfaces (Figure1). Finally, the reflectance contrast is given by dR 2 2 2 = r r / r (6) R re f − sample re f where r is the field reflection coefficient for the reference air hBN α-Al O , which is determined re f → → 2 3 by setting Mˆ 2 3 = 1ˆ. This procedure makes no perturbative assumptions about the index of refraction − of BP, and allows for generalization to thicker layers such as the bulk BP case. Materials 2020, 13, 5736 5 of 12

3. Results and Discussion Complex index of refraction for few-layer and bulk BP: The complex index of refraction for few-layer BP is then determined by fitting the reflectance contrast data [25] using non-linear least squares fitting ofMaterials the weight 2020, 13 (plasma, x FOR PEER frequencies) REVIEW parameters in the CKKA method with triangular basis functions;5 of 12 this method is referred to as CKKA+TMM [37,38]. The dR/R data were extracted from the literature using a plot plot digitizer digitizer with with high high accuracy accuracy [44]. [44 ].The The resulting resulting dR/R dR fits,/R fits, real real indices indices of refraction of refraction n andn andextinction extinction coefficients coefficients k, fork ,both for both x- and x- andy-polarized y-polarized excitation excitation for one for oneto five to fivelayers layers and andbulk bulk BP, are BP, aredisplayed displayed in Figure in Figure 2. We2. We note note the the high high quality quality of of the the dR/R dR /R fits fits (Figure (Figure 2a–e2a–e),), even small variationsvariations in dRdR/R/R are easily reproduced withwith CKKACKKA+TMM;+TMM; a high-resolution mesh of 300 triangular oscillators was chosenchosen to to fit fit these these data. data. We argueWe argue that thethatconstrained the constrainedKK method KK utilizedmethod hereutilized provides here credibilityprovides tocredibility the resulting to the complex resulting indices complex of refraction,indices of refraction, since the dispersive since the dispersive component component (n, or εr) is (n connected, or 𝜀) is directlyconnected via directly the KK via relations the KK to relations the absorptive to the componentabsorptive (componentk, or εi), which (k, or for 𝜀), atomically which for thin atomically films is thethin major films contributionis the major contribution to the dR/Rmeasurement to the dR/R measurement [10,29]. [10,29].

Figure 2. Complex index of refraction: results of fittin fittingg reflectance reflectance contrast dR/R dR/R data for 1–5 L and bulk BP, takentaken fromfrom Li.Li. et al. [[23]23] usingusing the constrainedconstrained Kramers–Kronig analysis (CKKA) method. First rowrow ( a(–ae–):e): polarization-selective polarization-selective reflectance reflectance contrast contrast dR/R( 1dR/RRsample ( 1−𝑅/Rre f ) data,⁄𝑅 with) constraineddata, with − Kramers–Kronigconstrained Kramers–Kronig analysis and analysis transfer matrixand transfer method matrix (CKKA method+TMM) (CKKA+TMM) fits. Secondrow fits. (Secondf–j): real row index (f– ofj): refractionreal index n extractedof refraction using n CKKAextracted+TMM using method. CKKA+TMM Third row method. (k–o): extinctionThird row coe (kffi–ocient): extinction k using CKKAcoefficient+TMM k using method. CKKA+TMM Fourth row method. (p–r): dRFourth/R, n, row k, for (p– bulkr): dR/R, (100 nm)n, k, BP.for bulk (100 nm) BP.

As it has been reported previously [[25],25], the reﬂectancereflectance contrast data for 1–5L BP (taken at 77 K) shows eithereither one one or twoor peakstwo withinpeaks thewithin experimental the experimental acquisition rangeacquisition of 0.75–2.4 range eV,corresponding of 0.75–2.4 eV, to bothcorresponding the intralayer to both exciton the intralayer at lower energies exciton at and lowe a blue-shiftedr energies and exciton a blue-shift that arisesed exciton due to that interlayer arises edueﬀects to interlayer similar to effects quantum similar wells to [quantum24]. We note wells that [24]. x- We and note y-polarized that x- and excitation y-polarized here excitation corresponds here tocorresponds polarization to alignedpolarization along aligned the AC along and ZZthe directionsAC and ZZ of directions the puckered of the few-layer puckered BP, few-layer respectively. BP, respectively. This puckering effect leads to significant in-plane anisotropy, resulting in a band structure for which the effective electron mass along the AC direction is five times larger than the ZZ direction [24]. As revealed both in the dR/R data, as well as in our extracted complex indices of refraction, sharp excitonic features are observed only in the AC (x) case in dR/R and k, with strongly dispersive lineshapes in n imposed on a relatively flat background which varies from 2–4 (Figure 2k–o). In the case of ZZ excitation, both the index n and extinction coefficient k are relatively featureless: the index is relatively flat between 2.5–3 from 1–2.2 eV in the cases of 2–5 L (Figure 2g–j, pink dashed curve),

Materials 2020, 13, 5736 6 of 12

This puckering effect leads to significant in-plane anisotropy, resulting in a band structure for which the effective electron mass along the AC direction is five times larger than the ZZ direction [24]. As revealed both in the dR/R data, as well as in our extracted complex indices of refraction, sharp excitonic features are observed only in the AC (x) case in dR/R and k, with strongly dispersive lineshapes in n imposed on a relatively flat background which varies from 2–4 (Figure2k–o). In the case of ZZ excitation, both the index n and extinction coefficient k are relatively featureless: the index is relatively flat between 2.5–3 from 1–2.2 eV in the cases of 2–5 L (Figure2g–j, pink dashed curve), while the extinction coefficient k increases from 0 to 4 between these energy ranges (Figure2l–o, pink dashed curve) . In fact, for both n and k, the results for ZZ excitation seem to constitute the background features in the AC case, indicating the presence of shared higher energy resonances which are isotropic. Thus, by examining the extinction coefficient k directly, which corresponds to absorption, we confirm previous statements in the literature that there exists a broad and increasing background absorption in few-layer BP towards higher energy, on which the strong and narrow excitonic resonances are super-imposed. The CKKA+TMM analysis used here is more general than the linearized models often used for the treatment of static absorption by atomically thin materials such as TMDs and few-layer BP [10,25]. Indeed, this model is used here to extract quantitatively meaningful results for the complex index of refraction, without resorting to approximations such as small optical path length for the encapsulation layer, minimal absorption in the BP layers or the neglect of the substrate reflection at the BP → α-Al2O3 interface. This generality allows us to also fit the 100 nm bulk BP sample (Figure2, fourth row). The dR /R data are no longer easily directly correlated with the absorption coefficient k, due to the non-negligible thickness of bulk BP, as well as the interference effects with the hBN encapsulating layer. High quality fits of the dR/R data reveal again large anisotropy resulting in birefringence of bulk BP, with a modulation of the index of refraction around 0.32 eV in the AC case, which is not observed in the ZZ excitation case. For the extinction coefficient k, in the AC case, the absorption shows an onset around the band edge at 0.32 eV, but also shows absorption at lower energies likely due to the Drude response from free carrier excitation [24]. Surprisingly, for the ZZ excitation case, the absorption also increases steadily towards higher energies, without any dispersion modulation around the band edge. We also note that in this energy range, the extracted index of refraction for bulk BP is below 1. Sub-unity indices of refraction are commonly reported in noble metal materials such as Ag and Au, where there is an interplay between interband absorption and the plasmonic response [45–47] as well as in Al thin films at wavelengths shorter than 660 nm [48]. Indeed, a zero-crossing of the real part of the dielectric function εr is required to satisfy the conditions for a localized surface plasmon resonance (LSPR) [49]; the plasmonic nature of BP is currently under investigation, due its novel features such as hyperbolic plasmonics arising from the large ratio between the AC and ZZ direction effective electron mass [24]. Systematics: In this section, we remark on the sensitivity of the CKKA+TMM analysis for the extraction of the complex indices of refraction to variations in parameters such as substrate and encapsulation layer refractive indices, as well as fixed fitting parameters. Previous reports involving the utilization of the CKKA method have commented on the need for incorporating information about low and high energy extremes of the dielectric functions outside of the experimental acquisition range. For instance, Li et al. [10] assumed during their studies of TMD monolayers that at high energies (>3 eV), the bulk optical properties also contributed to the monolayer optical response; absorption resonances were included up to 30 eV. No such assumptions are made in our determination of n,k for few-layer BP or the bulk BP. However, in Figure3, a convergence of the fitting results for 1L-x BP is shown as a function of the buffer energy that is included in the fits. More specifically, the experimental data were collected from 0.75–2.4 eV; additionally, we examined the convergence of the fit solution as a function of how far the oscillator energy mesh extended beyond these extremal values, i.e., h i E E E , E + E . Although the imaginary part of the CKKA basis triangular mesh ∈ data,i − bu f data, f bu f Materials 2020, 13, 5736 7 of 12 dielectric function is well localized in energy, the real part exhibits long tails away from the oscillator center; thus, by extending our oscillator energy range, we can simulate how these low and high energy contributions increase the quality of fit. This buffer energy was tested from 0 to 0.3 eV; it is shown (Figure3, center, inset) that increasing the bu ffer energy from 0 to 0.1 eV reduces the sum of the residuals by nearly two orders of magnitude, with generally no enhancement beyond 0.1 eV. This convergence can be seen in the results for n and k. However, it is noted that at energies greater than 2.4 eV that Materials 2020, 13, x FOR PEER REVIEW 7 of 12 the real index of refraction n varies considerably for different buffer energies; random variations areobserved also observed in this energy in this range energy for slight range variations for slight in variations the number in theof energy number mesh of energy points mesh(not shown). points (notWe believe shown) .that We this believe lack that of convergence this lack of convergence arises because arises of becausethe lack ofof theconstr lackaints of constraints on the dielectric on the dielectricfunction we function have imposed we have on imposed energies on higher energies than higher 2.5 eV; than this 2.5 aspect eV; this may aspect be potentially may be potentially improved improvedby using the by usinghigher the energy higher absorption energy absorption data for databulk forBP, bulk in a BP,similar in a similarmanner manner as done as with done TMD with TMDmonolayers. monolayers.

(a) (b) (c)

Figure 3.3.Systematic Systematic variation variation in CKKA in CKKA+TMM+TMM parameters. parameters. Left (a) andLeft center (a) (andb) plots: center demonstration (b) plots: ofdemonstration convergence of of convergenc CKKA+TMMe of fitsCKKA+TMM for x-polarized fits for excitation x-polarized of 1L excitation BP as a functionof 1L BP ofas mesha function energy of bumeshffer. energy This bu buffer.ffer is definedThis buffer as the is defined energy range as the extending energy range beyond extending the experimental beyond the data experimental acquisition rangedata acquisition used in the range CKKA used+TMM in the fit.CKKA+TMM (a): real index fit. (a of): real refraction index of n refraction as a function n as ofa function buffer energy of buffer in eVenergy (see in legend eV (see in legend plot on in right). plot on (b ):right). extinction (b): extinction coefficient coefficient k as a function k as a function of buffer of energy. buffer Insetenergy. of

(Insetb): log of10 (bof): log the10 residuals of the residuals of the dR of/ Rthe fits dR/R as a fits function as a fu ofnction buffer of energy. buffer energy. Residuals Residuals are defined are defined here as P 2 here x f itasj ∑x𝑥data j −𝑥.(c): variation . (c): ofvariation the index of ofthe refraction index of for refrac the 15tion nm for hexagonal the 15 nm boron hexagonal nitride (hBN)boron , − ,, , encapsulatingnitride (hBN) encapsulating layer. hBN(E), layer. hBN || hBN(E),and hBN hBN iso correspond|| and hBN to iso using correspond the Sellmeier to using equation the Sellmeier form of

theequation index form of refraction of the index for in-plane of refraction hBN, for a constantin-plane valuehBN, ofa constantnhBN = 2.108 value [ 33of], 𝑛 and a = constant2.108 [33], value and

na hBNconstant= 1.86 value [34], respectively.𝑛 = 1.86 Solid[34], linesrespectively. correspond Solid to lines n of BP, correspond and dashed to linesn of BP, correspond and dashed to k oflines BP, ascorrespond indicated to by k solid of BP, and as dashedindicated arrows. by solid and dashed arrows.

The dependence of the results for 1L-x BP were also studied as a function of the hBN refractive index. ItIt isis knownknown that that exfoliated exfoliated hBN hBN is birefringent,is birefringent, but but estimates estimates of the of forthe in-plane for in-plane and out-of-planeand out-of- indicesplane indices of refraction of refraction have have varied varied significantly significan betweently between reports reports [33– [33–35].35]. For For the the results results displayed displayed in Figurein Figure2, as 2, well as well as Figure as Figure3, a Sellmeier3, a Sellmeier equation equation is used is used that describesthat describes the index the index of refraction of refraction of the of in-planethe in-plane index, index, which which ranges ranges from from 2.25 2.25 to to 2.1 2.1 from from 400–1600 400–1600 nm nm [33 [33].]. The The fittingfitting procedureprocedure waswas also performed for a constant index index nn == 2.108,2.108, or or the the average average of of the the in-plane in-plane index, index, and and n n= =1.86,1.86, a acommonly commonly used used value value for forisotropic isotropic BN [33–35] BN [33– (Figure35] (Figure 3a,b).3a,b). Using Using the constant the constant in-plane in-plane value valueyields yieldsa generally a generally lower value lower than value the than Sellmeier the Sellmeier case, and case, using and the using isotropic the isotropic value leads value to leads much to larger much largerestimates estimates of n; the of n variations; the variations in k inarek notare notas significant as significant (Figure (Figure 3c).3c). We We believe believe that that the the use ofof thethe functional Sellmeier equation form is appropriate; the measurement resulting in that formform utilizedutilized confocal oblique incidence ellipsometry to capture both the in-plane and out-of-planeout-of-plane indices from 400–1600 nm [[33].33]. Calculation of reflectance contrast of BP on SiO /Si substrates: One immediately apparent use for the Calculation of reflectance contrast of BP on SiO22/Si substrates: One immediately apparent use for the complex indices of refraction ofof BPBP determineddetermined inin thisthis paperpaper usingusing CKKACKKA+TMM+TMM is the simulation of the optical properties of complex multi-layered devices incorporating BP. During the characterization and development of novel optical devices that incorporate exfoliated materials such as TMDs and BP, one step in the standard procedure for exfoliation involves the isolation of small flakes by thickness. This identification often involves measuring the reflectance contrast over the visible spectrum, i.e., examining a microscope image by color contrast (by eye or CCD) [50]. Thus, maximizing the color contrast between the active layer (TMD or BP) and the substrate is essential for this exfoliation procedure. If the reflectance contrast curves can be predicted ahead of time given knowledge of the complex indices of refraction of the TMD/BP and the substrate, the thickness of the active layer can be determined with certainty.

Materials 2020, 13, 5736 8 of 12 and development of novel optical devices that incorporate exfoliated materials such as TMDs and BP, one step in the standard procedure for exfoliation involves the isolation of small flakes by thickness. This identification often involves measuring the reflectance contrast over the visible spectrum, i.e., examining a microscope image by color contrast (by eye or CCD) [50]. Thus, maximizing the color contrast between the active layer (TMD or BP) and the substrate is essential for this exfoliation procedure. If the reflectance contrast curves can be predicted ahead of time given knowledge of the complexMaterials 2020 indices, 13, x FOR of refraction PEER REVIEW of the TMD/BP and the substrate, the thickness of the active layer can8 ofbe 12 determined with certainty. InIn TMDTMD systems,systems, one one common common substrate substrate employed employed for for high high color color contrast contrast is 90–120is 90–120 nm nm of SiO of SiO2 on2 aon Si a substrate Si substrate [28]. [28]. Here, Here, we employedwe employed the TMM the TMM to calculate to calculate the expected the expected reflectance reflectance contrast contrast of one toof threeone to layers three oflayers BP on of thatBP on same that substrate, same substrate, using the using complex the comple dielectricx dielectric constants constants determined determined in this articlein this viaarticle CKKA via+ TMM.CKKA+TMM. Thus, we Thus, have we provided have provided a useful reference a useful forreference optical physicistsfor optical attempting physicists toattempting incorporate to incorporate BP in their devices.BP in their The devices. results The of theresults calculation of the calculation are shown are in Figureshown4 in. ItFigure becomes 4. It immediatelybecomes immediately clear that thisclear standard that this substratestandard doessubstr notate provide does not high provide reflectance high reflectance contrast, at contrast, least not asat highleast asnot using as high sapphire as using as asapphire substrate as(Figure a substrate2). This (Fig lackure 2). of reflectanceThis lack of contrast reflectance is due contrast to the is relative due to similaritiesthe relative of similarities the real parts of the of the real index parts of of refraction the index for of Si refraction and 1–3 L for BP, Si with and n 1–3 ~ 3–4 L fromBP, with 0.8 to n 2.5~ 3–4 eV (Figurefrom 0.82). to However, 2.5 eV the(Figure contrast 2). However, between the the x- andcontrast y-polarized between excitation the x- and reflectance y-polarized contrast excitation allows onereflectance to clearly contrast locate allows sharp features one to clearly in the opticallocate sharp spectrum, features which in th cane optical then be spectrum, used for which layer thickness can then determination.be used for layer For thickness easy thickness determination. determination, For easy we suggestthickness utilizing determination, either sapphire we suggest as a substrate, utilizing oreither polarization-dependent sapphire as a substrate, reflectance or polari contrast.zation-dependent reflectance contrast.

Figure 4.4. ReﬂectanceReflectance contrastcontrast (dR (dR/R)/R) calculated calculated using usin n,kg n,k displayed displayed in Figurein Figure2 between 2 between (1L ((1La), 2L(a), (b2L),

and(b), 3Land ( c3L), BP (c), on BP 90 on nm 90 SiO nm2/ Si)SiO and2/Si) 90 and nm 90 SiO nm2/Si. SiO Solid2/Si. blue,Solid dashedblue, dashed blue and blue solid and orange solid orange curves correspondcurves correspond to dR/R to for dR/R the x-polarization,for the x-polarization, y-polarization y-polarization excitation excitation conditions conditions and the di andﬀerence the betweendifference the between two, respectively. the two, respectively.

Discussion ofof study limitations, andand comparisons toto thethe literature:: Some of the limitations of this study have been discussed discussed above, above, and and we we re-iterate re-iterate in inmore more detail detail here. here. First, First, as discussed as discussed in detail in detail by Li by et Lial. et[10] al. [and10] andKuzmenko Kuzmenko [37], [37 both], both the the standard standard Kramers–Kronig Kramers–Kronig method method and and the the CKKA method utilized in this article are sensitivesensitive to thethe opticaloptical responseresponse ofof thethe systemsystem outsideoutside ofof thethe experimentalexperimental acquisition range.range. The The standard standard KK KK approach approach requires requires that thethat absorptive the absorptive optical responseoptical response is localized is overlocalized an energy over an range energy smaller range thansmaller the than experimental the experimental energy energy range; range; while thewhile CKKA the CKKA method method does notdoes explicitly not explicitly require require this, this, we showedwe showed in thein theSystematics Systematicssection section of of this this article article that that thethe extractedextracted complex index ofof refractionrefraction hashas highhigh uncertaintyuncertainty atat thethe edgesedges ofof the the data data acquisition acquisition range. range. TheThese se uncertainties would be greatly reduced by incorporatingincorporating both higher and lower energy data that could bebe acquiredacquired via via electron electron energy-loss energy-loss measurements measurements and and UV-XUV UV-XUV absorption absorption measurements measurements [41], as[41], well as well as long as long wavelength wavelength FTIR FTIR spectroscopy. spectroscopy We note. We that note it that is non-trivial it is non-trivial to incorporate to incorporate the results the ofresults those of higher those/lower higher/lower energy experiments energy experiment into CKKAs into+TMM, CKKA+TMM, since those since experiments those experiments span over highly span disparateover highly energy disparate ranges energy and apply ranges to theand bulk apply crystal to the black bulk phosphorous, crystal black butphosphorous, point the reader but point towards the referencedreader towards literature referenced for future literatur studiese for [51 future–54]. studies [51–54]. Second, bothboth thethe KKKK and and CKKA CKKA methods methods require require precise precise characterization characterization of theof the thicknesses thicknesses of the of substratethe substrate layers layers with with nanometer nanomete precision,r precision, and knowledgeand knowledge of the of indices the indices of refraction of refraction for those forlayers. those layers. In this article, the hBN layer thickness has been taken as approximately 15 nm, as given by Li et al. [25]. We suggest that a more consistent approach would involve multiple reflectance contrast measurements of BP with varying hBN encapsulation layer thicknesses, with thicknesses of all layers characterized by atomic force microscopy. The same concern applies to the extraction of the complex index of the bulk BP described in this article; this thickness was taken to be equal to approximately 100 nm, as given by Li et al. [25], but the uncertainty on this thickness is unknown to us. Additionally, we are uncertain of the crystallinity of the hBN encapsulation layer, as discussed in the Systematics section of this article, but have provided systematic variations over the possible cases. In the context of the literature regarding the optical characterization of TMDs and few-layer BP, and more generally exfoliated materials, this article takes a similar approach to that of Li et al. [10],

Materials 2020, 13, 5736 9 of 12

In this article, the hBN layer thickness has been taken as approximately 15 nm, as given by Li et al. [25]. We suggest that a more consistent approach would involve multiple reﬂectance contrast measurements of BP with varying hBN encapsulation layer thicknesses, with thicknesses of all layers characterized by atomic force microscopy. The same concern applies to the extraction of the complex index of the bulk BP described in this article; this thickness was taken to be equal to approximately 100 nm, as given by Li et al. [25], but the uncertainty on this thickness is unknown to us. Additionally, we are uncertain of the crystallinity of the hBN encapsulation layer, as discussed in the Systematics section of this article, but have provided systematic variations over the possible cases. In the context of the literature regarding the optical characterization of TMDs and few-layer BP, and more generally exfoliated materials, this article takes a similar approach to that of Li et al. [10], in which the complex dielectric functions of monolayer TMDs were determined via CKKA+TMM. The analysis provided here clearly separates the dispersive (real) and absorptive (imaginary) parts of the optical dielectric constants of few-layer and bulk BP, which is not obvious in reﬂectance contrast data.

4. Conclusions For the estimate of the index of the refraction of black phosphorus, we have demonstrated that the constrained KK analysis combined with a general transfer matrix method (CKKA+TMM) that takes into account finite layer optical path lengths can be used to extract the entire complex index of the refraction of 1–5 L BP and bulk BP from reflectance contrast data. It is confirmed that the excitonic peaks correspond to peaks in the extinction coefficient k, and that a significant absorptive background that increases towards higher energies exists. In contrast, the real index of refraction is relatively flat with superimposed dispersive features located at the excitonic resonances. The determination of the full complex index of refraction en of few-layer BP is significant: these results may be incorporated into the development of novel optical devices such as responsive photonic crystals and distributed Bragg reflectors (DBRs) [55–58], plasmonic nanostructures [59] and on-chip waveguides, all of which require knowledge of both the dispersive and absorptive optical properties. We calculated the reflectance contrast of 1–3 L BP on a SiO2/Si substrate, providing a useful reference for physicists working to determine thicknesses of exfoliated BP. We suggest further improvements to this extractive method that would involve the experimental acquisition of both reflectance and transmission contrast simultaneously, as well as constraining the low and high energy ranges in the CKKA+TMM analysis with optical properties deduced from FTIR and UV absorption.

Author Contributions: Conceptualization, A.M.R., G.M.P., S.D.C., F.S. and E.C.; methodology, validation and formal analysis, A.M.R., F.S. and E.C.; writing—original draft preparation, A.M.R. and F.S.; writing—review and editing, A.M.R., G.M.P., S.D.C., F.S. and E.C. All authors have read and agreed to the published version of the manuscript. Funding: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. [816313]). E.C. acknowledges financial support from MIUR PRIN aSTAR, Grant No. 2017RKWTMY and from the European Research Council through the MSCA-ITN SMART-X (GA 860553). G.M.P. acknowledges the financial support from Fondazione Cariplo, grant no. 2018-0979. S.D.C. acknowledges financial support from MIUR through the PRIN 2017 Programme (Prot. 20172H2SC4). Conflicts of Interest: The authors declare no conflict of interest.

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